Subgroups — #33094145

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620254	C₂	1	1	yes	maximal, characteristic
2	#22618868	C₂	1	1	yes	maximal, characteristic, complemented
3	#22618648	C₂	1	1	yes	maximal, characteristic
4	#22610911	C₂	1	1	yes	maximal, characteristic, complemented
5	#22610762	C₂	1	1	yes	maximal, characteristic
6	#22610640	C₂	1	2	yes	maximal, complemented
7	#22610633	C₂	1	2	yes	maximal, complemented
8	#22609774	C₂	1	2	yes	maximal, complemented
9	#22609767	C₂	1	2	yes	maximal, complemented
10	#22609617	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556599	C₂²	1	2	yes	—
13	#17989144	C₂²	1	1	yes	characteristic
14	#17989137	C₂²	1	1	yes	characteristic
15	#17988942	C₄	1	4	yes	complemented
16	#17988938	C₄	1	2	yes	complemented
17	#17988934	C₄	1	2	yes	complemented
18	#17988716	—	2	4	no	—
19	#17988657	—	2	4	no	—
20	#17988615	C₂²	1	2	yes	—
21	C₂×#14556517	C₂²	1	2	yes	—
22	C₂×#14556517	C₂²	1	2	yes	—
23	#17988584	C₂²	1	2	yes	—
24	#17988584	C₂²	1	2	yes	—
25	#17988555	C₂²	1	2	yes	—
26	#17988545	C₂²	1	2	yes	—
27	C₂×#14556502	C₂²	1	1	yes	characteristic
28	C₂×#14556501	C₂²	1	1	yes	characteristic
29	C₂×#14556501	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
32	#17988433	C₂²	1	1	yes	characteristic
33	#17988348	C₂²	1	2	yes	—
34	#17988348	C₂²	1	2	yes	—
35	#423×C₇₃	C₂²	1	1	yes	characteristic
36	#423×C₇₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
38	#341×C₇₃	C₂²	1	2	yes	—
39	#341×C₇₃	C₂²	1	2	yes	—
40	C₂×#11924028	C₂³	1	2	yes	—
41	#14556599	—	2	4	no	—
42	#14556599	—	2	4	no	—
43	C₂×#11924011	—	2	2	no	—
44	C₂²×#10596831	C₂³	1	1	yes	characteristic
45	C₂²×#10596831	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₇₃	—	2	2	no	—
47	#14556591	C₂×C₄	1	1	yes	characteristic
48	#14556591	C₂×C₄	1	2	yes	—
49	#14556588	C₂×C₄	1	1	yes	characteristic
50	#14556583	C₂×C₄	1	2	yes	—
51	#14556579	C₂×C₄	1	2	yes	—
52	#14556570	—	2	2	no	—
53	#14556565	—	2	2	no	—
54	C₄×Dic₁₄₆	—	2	2	no	—
55	C₄×D₂₉₂	—	2	2	no	—
56	#14556550	C₂³	1	2	yes	—
57	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
59	#14556520	—	2	4	no	—
60	#14556517	—	2	4	no	—
61	#14556517	—	2	4	no	—
62	#14556516	C₂×C₄	1	4	yes	—
63	#14556516	C₂×C₄	1	4	yes	—
64	#14556513	C₂×C₄	1	4	yes	—
65	#14556513	C₂×C₄	1	4	yes	—
66	#14556503	—	2	4	no	—
67	#14556502	D₄	1	4	yes	—
68	#14556501	—	2	2	no	—
69	#14556501	—	2	2	no	—
70	#14556501	—	2	2	no	—
71	C₈×Dic₇₃	—	2	2	no	—
72	C₈×Dic₇₃	—	2	2	no	—
73	C₈×Dic₇₃	—	2	2	no	—
74	#14556498	—	2	2	no	—
75	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
76	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₇₃	—	2	2	no	—
81	C₄⋊C₈×C₇₃	—	2	2	no	—
82	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
84	#11924028	—	2	2	no	—
85	#11924028	—	2	2	no	—
86	#11924028	—	2	2	no	—
87	#11924028	—	2	2	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
90	C₂×D₂₉₂	—	2	2	no	—
91	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
92	C₂×C₂²⋊D₇₃	—	2	2	no	—
93	C₂²×Dic₇₃	—	2	1	no	—
94	C₂²×Dic₇₃	—	2	1	no	—
95	C₂×C₄×D₇₃	—	2	2	no	—
96	#11924011	—	4	4	no	—
97	C₂×#10596831	—	2	1	no	—
98	C₂×#10596831	—	2	1	no	—
99	C₂×#10596831	C₂×D₄	1	2	yes	—
100	C₂×#10596831	C₂×D₄	1	2	yes	—
101	C₂×#10596831	#41	1	2	yes	—
102	C₂×#10596831	#41	1	2	yes	—
103	C₈×D₇₃	—	4	4	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	C₄⋊Dic₇₃	—	2	2	no	—
106	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
107	#11924007	—	2	4	no	—
108	C₂²⋊Dic₇₃	—	2	2	no	—
109	#11924005	—	2	4	no	—
110	C₄×Dic₇₃	—	2	2	no	—
111	C₄×Dic₇₃	—	2	1	no	—
112	C₄×Dic₇₃	—	2	1	no	—
113	C₄×Dic₇₃	#41	1	2	yes	—
114	C₄×Dic₇₃	#41	1	2	yes	—
115	C₂²×C₄×C₇₃	#41	1	2	yes	—
116	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₇₃	—	2	2	no	—
118	M₁₆×C₇₃	—	2	2	no	—
119	M₁₆×C₇₃	—	2	2	no	—
120	M₁₆×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	2	1	no	—
122	C₂×C₈×C₇₃	—	2	1	no	—
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	1	no	—
125	C₂×C₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	—	2	2	no	—
127	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
129	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₇₃	C₂×D₄	1	2	yes	—
132	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₇₃	—	2	2	no	—
134	C₂×C₄×C₇₃	—	4	2	no	—
135	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₇₃	—	4	2	no	—
144	C₂²⋊D₇₃	—	4	2	no	—
145	C₂×Dic₇₃	—	2	4	no	—
146	C₂×Dic₇₃	—	2	2	no	—
147	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	D₂₉₂	—	2	2	no	—
152	D₂₉₂	—	2	2	no	—
153	D₂₉₂	—	8	2	no	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	Dic₁₄₆	—	2	2	no	—
157	Dic₁₄₆	—	2	2	no	—
158	Dic₁₄₆	—	8	2	no	—
159	C₈×C₇₃	—	2	2	no	—
160	C₈×C₇₃	—	2	2	no	—
161	C₈×C₇₃	—	4	2	no	—
162	C₈×C₇₃	—	4	2	no	—
163	#10596831	—	2	2	no	—
164	#10596831	#124	1	4	yes	—
165	C₂²×C₇₃	—	2	2	no	—
166	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₇₃	#529	1	1	yes	characteristic
168	C₂²×C₇₃	#529	1	1	yes	characteristic
169	C₂×D₇₃	—	4	4	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₄×C₇₃	—	2	2	no	—
172	C₄×C₇₃	—	2	2	no	—
173	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
175	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
176	C₄×C₇₃	#532	1	1	yes	characteristic
177	Dic₇₃	—	4	2	no	—
178	Dic₇₃	—	4	2	no	—
179	Dic₇₃	—	4	4	no	—
180	Dic₇₃	—	4	2	no	—
181	#21659	—	73	1	no	maximal, Sylow
182	C₂×C₇₃	—	2	1	no	—
183	C₂×C₇₃	—	2	1	no	—
184	C₂×C₇₃	C₄×#124	1	1	yes	characteristic
185	C₂×C₇₃	#2957	1	2	yes	—
186	D₇₃	—	8	2	no	—
187	#3131	—	73	1	no	—
188	#2961	—	73	1	no	—
189	#2932	—	73	1	no	—
190	#1995	—	73	1	no	—
191	#1963	—	73	1	no	—
192	#1938	—	73	2	no	—
193	#1937	—	73	2	no	—
194	#1837	D₇₃	1	1	yes	characteristic, complemented
195	#1765	—	73	2	no	—
196	#1764	—	73	2	no	—
197	#1726	—	73	1	no	—
198	C₇₃	#21659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#131	—	73	2	no	—
200	#533	—	73	1	no	—
201	#532	—	73	1	no	—
202	#488	—	73	4	no	—
203	#487	—	73	2	no	—
204	#486	—	73	2	no	—
205	#446	—	146	4	no	—
206	#435	—	146	4	no	—
207	#424	—	73	2	no	—
208	#423	C₂×D₇₃	1	1	yes	characteristic
209	#423	C₂×D₇₃	1	1	yes	characteristic
210	C₂×#104	—	73	2	no	—
211	C₂×#104	—	73	2	no	—
212	#416	—	73	2	no	—
213	#416	—	73	2	no	—
214	#408	—	73	2	no	—
215	#405	—	73	2	no	—
216	C₂×#97	—	73	1	no	—
217	C₂×#97	—	73	1	no	—
218	C₂×#97	—	73	1	no	—
219	C₂×C₄×C₈	—	73	1	no	—
220	C₂×C₄×C₈	—	73	1	no	—
221	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
222	#376	—	73	1	no	—
223	#341	C₂×D₇₃	1	2	yes	—
224	#341	C₂×D₇₃	1	2	yes	—
225	#339	—	73	2	no	—
226	#339	—	73	2	no	—
227	C₂×#41	—	73	2	no	—
228	#131	—	146	4	no	—
229	#131	—	146	4	no	—
230	C₂×M₁₆	—	146	2	no	—
231	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
232	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
233	C₂²×C₈	—	73	1	no	—
234	C₂²×C₈	—	73	1	no	—
235	C₂²×C₈	—	146	2	no	—
236	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₄⋊Q₈	—	73	2	no	—
239	C₄⋊Q₈	—	73	1	no	—
240	#127	—	73	1	no	—
241	#125	—	73	2	no	—
242	#124	—	73	2	no	—
243	C₂²⋊Q₈	—	146	2	no	—
244	#121	—	146	2	no	—
245	C₄×Q₈	—	146	2	no	—
246	C₄×D₄	—	146	2	no	—
247	#117	—	73	2	no	—
248	C₂×C₄²	—	73	1	no	—
249	C₂×C₄²	—	73	1	no	—
250	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	146	4	no	—
254	#104	—	146	4	no	—
255	#104	—	146	4	no	—
256	#103	—	73	4	no	—
257	#103	—	73	4	no	—
258	#102	—	73	4	no	—
259	#102	—	73	4	no	—
260	C₂²⋊C₈	—	146	4	no	—
261	#97	—	73	4	no	—
262	#97	—	146	2	no	—
263	#97	—	146	2	no	—
264	#97	—	146	2	no	—
265	C₄×C₈	—	146	2	no	—
266	C₄×C₈	—	146	2	no	—
267	C₄×C₈	—	146	2	no	—
268	C₄×C₈	C₄×D₇₃	1	2	yes	—
269	C₄×C₈	C₄×D₇₃	1	2	yes	—
270	#95	—	146	2	no	—
271	#41	—	146	2	no	—
272	#41	—	146	2	no	—
273	#41	—	146	2	no	—
274	#41	—	146	2	no	—
275	C₂×Q₈	—	73	2	no	—
276	C₂×Q₈	—	146	2	no	—
277	C₂×D₄	—	73	2	no	—
278	C₂×D₄	—	146	2	no	—
279	C₂×D₄	—	146	2	no	—
280	C₂²×C₄	—	146	1	no	—
281	C₂²×C₄	—	146	1	no	—
282	C₂²×C₄	—	146	2	no	—
283	C₂²×C₄	#11924029	1	2	yes	—
284	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	292	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	292	4	no	—
297	C₂×C₈	—	73	2	no	—
298	C₂×C₈	—	73	2	no	—
299	C₂×C₈	—	73	2	no	—
300	C₂×C₈	—	73	2	no	—
301	C₂×C₈	—	146	1	no	—
302	C₂×C₈	—	146	1	no	—
303	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
304	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
305	C₄⋊C₄	—	73	4	no	—
306	C₄⋊C₄	—	146	4	no	—
307	C₄⋊C₄	—	146	2	no	—
308	C₄⋊C₄	—	146	2	no	—
309	C₂²⋊C₄	—	146	2	no	—
310	C₂²⋊C₄	—	146	4	no	—
311	C₄²	—	73	2	no	—
312	C₄²	—	73	2	no	—
313	C₄²	—	146	2	no	—
314	C₄²	—	146	1	no	—
315	C₄²	—	146	1	no	—
316	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
318	C₄²	D₄×D₇₃	1	2	yes	—
319	C₂³	—	146	2	no	—
320	C₂³	C₂×#11924029	1	1	yes	characteristic
321	Q₈	—	584	2	no	—
322	Q₈	—	146	2	no	—
323	Q₈	—	146	2	no	—
324	D₄	—	292	2	no	—
325	D₄	—	292	2	no	—
326	D₄	—	584	2	no	—
327	D₄	—	146	2	no	—
328	D₄	—	146	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	292	2	no	—
331	C₂×C₄	—	292	2	no	—
332	C₂×C₄	—	292	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	4	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	#14556553	1	2	yes	—
339	C₂×C₄	#14556553	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
345	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	73	4	no	—
351	C₈	—	146	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	292	4	no	—
354	C₂²	—	292	2	no	—
355	C₂²	C₂×#14556553	1	1	yes	characteristic
356	C₂²	#17989122	1	1	yes	characteristic
357	C₂²	#17989122	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	292	2	no	—
361	C₄	—	292	2	no	—
362	C₄	—	292	4	no	—
363	C₄	—	292	2	no	—
364	C₄	#17989055	1	1	yes	characteristic
365	C₄	#17989055	1	1	yes	characteristic
366	C₄	#124×D₇₃	1	1	yes	characteristic
367	C₄	#17989136	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	584	2	no	—
371	C₂	#22612470	1	1	yes	characteristic
372	C₂	#22618847	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1837	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1837	D₇₃
hypocenter	C₇₃	#21659
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22618868	C₂	semidirect
4. #22610911	C₂	semidirect
6. #22610640	C₂	semidirect
7. #22610633	C₂	semidirect
8. #22609774	C₂	semidirect
9. #22609767	C₂	semidirect
10. #22609617	C₂	semidirect
11. C₇₃×#1837	C₂	semidirect
15. #17988942	C₄	semidirect
16. #17988938	C₄	semidirect
17. #17988934	C₄	semidirect
194. #1837	D₇₃	semidirect
198. C₇₃	#21659	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22620254	2
#22618868	2
#22618648	2
#22610911	2
#22610762	2
#22610640	2
#22610633	2
#22609774	2
#22609767	2
#22609617	2
C₇₃×#1837	2
#21659	73